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Mirrors > Home > HSE Home > Th. List > dmdi2 | Structured version Visualization version GIF version |
Description: Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dmdi2 | ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝐶 ∩ 𝐴) ∨ℋ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmdi 32232 | . 2 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → ((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵))) | |
2 | eqimss2 4038 | . 2 ⊢ (((𝐶 ∩ 𝐴) ∨ℋ 𝐵) = (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) → (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝐶 ∩ 𝐴) ∨ℋ 𝐵)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ* 𝐵 ∧ 𝐵 ⊆ 𝐶)) → (𝐶 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝐶 ∩ 𝐴) ∨ℋ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∩ cin 3945 ⊆ wss 3946 class class class wbr 5145 (class class class)co 7416 Cℋ cch 30859 ∨ℋ chj 30863 𝑀ℋ* cdmd 30897 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-iota 6498 df-fv 6554 df-ov 7419 df-dmd 32211 |
This theorem is referenced by: (None) |
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